[
I5s]
THE INTRINSIC RATE OF NATURAL INCREASE
OF AN INSECT POPULATION
BY L. C. BIRCH*, Zoology Department, University of Sydney
CONTENTS
PAGE
PAGE
.
.
.
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.
.
I.
INTRODUCTION
2.
BIOLOGICAL SIGNIFICANCE OF THE INTRINSIC RATE
OF NATURAL
3.
CALCULATION
NATURAL
(a)
(b)
(c)
(d)
INCREASE
OF
INCREASE
THE
.
.
.
.
Experimental data required .
.
The net reproduction rate
The mean length of a generation
.
The calculation of 'r' .
R.
I6
4.
THE STABLE AGE DISTRIBUTION.
.
20
5.
THE INSTANTANEOUS BIRTH-RATE AND DEATH-RATE
2I
6.
THE EFFECT OF TEMPERATURE ON
.
'r'
.
2I
.
.
.
.
I7
7. DISCUSSION
.
.
.
.
.
.
.2
.
.
.
.
I7
I8
8.
SUMMARY
.
.
.
.
.
.
.
9.
ACKNOWLEDGEMENTS
.
.
.
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.
25
.
.
.
.
.
26
RATE
INTRINSIC
.
.
.
15
OF
.
.
I8
.
.
'9
i. INTRODUCTION
The intrinsic rate of increase is a basic parameter
which an ecologist may wish to establish for an insect
population. We define it as the rate of increase per
head under specified physical conditions, in an unlimited environment where the effects of increasing
density do not need to be considered. The growth of
such a population is by definition exponential. Many
authors, including Malthus and Darwin, have been
concerned with this and related concepts, but there
has been no general agreement in recent times on
definitions. Chapman (I93i) referred to it as 'biotic
potential', and although he does state in one place
that biotic potential should in some way combine
fecundity rate, sex ratio and survival rate, he never
precisely defined this expression. Stanley (I 946) discussed a somewhat similar concept which he called
the 'environmental index'. This gives a measure of
the relative suitability of different environments, but
it does not give the actual rate of increase of the insect
under these different conditions. An index for the
possible rate of increase under different physical conditions would at the same time provide a measure of
the relative suitability of different environments.
Birch (I 945c) attempted to provide this in an index
comnbiningthe total number of eggs laid, the survival
rate of immature stages, the rate of development and
the sex ratio. This was done when the author was
unaware of the relevance of cognate studies in human
demography. A sounder approach to insect populations based on demographic procedures is now
REFERENCES
.
.
2
25
suggested in this paper. The development of this
branch of population mathematics is principally due
to A. J. Lotka. From the point of view of the biologist,
convenient summaries of his fundamental contributions to this subject will be found in Lotka (I925,
Chapter 9; I939 and I945). A numerical example of
the application of Lotka's methods in the case of
a human population will be found in Dublin & Lotka
The parameter which Lotka has developed
(I925).
for human populations, and which he has variously
called the 'true' or 'inherent' or 'intrinsic' rate of
natural increase, has obvious application to populations of animals besides the human species. The first
determination of the intrinsic rate of increase of an
animal other than man was made by Leslie & Ranson
(I940). They calculated the 'true rate of natural
increase' of the vole, Microtus agrestis, from agespecific rates of fecundity and mortality determined
under laboratory conditions. With the use of matrices
Leslie has extended these methods and, as an example,
calculated the true rate of natural increase of the
brown rat, Rattus norvegicus(Leslie,
1945).
The
author is much indebted to Mr Leslie for having
drawn his attention to the possible application of
actuarial procedures to insect populations. He has
been completely dependent upon him for the methods
of calculation used in this paper.
Before proceeding to discuss the reasons for the
particular terminology adopted in this paper, it is
necessary first to consider the true nature of the
parameter with which we are concerned.
* This investigation was carried out at the Bureau of Animal Population, Oxford University, during the tenure of
an overseas senior research scholarship from the Australian Science & Industry Endowment Fund.
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I6
The intrinsicrate of natural increaseof an insectpopulation
z. BIOLOGICAL SIGNIFICANCE OF THE
INTRINSIC RATE OF NATURAL INCREASE
The intrinsic rate of increase is best defined as the
constant' r 'in the differential equation for population
increase in an unlimited environment,
dN/dt = rN,
or in the integrated form Nt = N0ert,
where No = number of animals at time zero,
Nt = number of animals at time t,
r = infinitesimal rate of increase.
The exponent r is the difference between the birthrate (b) and the death-rate (d) in the population
(r = b - d). In some circumstances it may be more
useful to know the finite rate of increase, i.e. the
number of times the population multiplies in a unit
of time. Thus, in a population which is increasing
exponentially, if there are Nt individuals at time
t then in one unit of time later the ratio
Nt+1
Nt
-e
r
= antiloge r = A.
Hence the finite rate of increase (A) is the natural
antilogarithm of the intrinsic (infinitesimal) rate of
increase.
Any statement about the rate of increase of a
population is incomplete without reference to the age
distribution of that population, unless every female
in it happens to be producing offspring at the same
rate at all ages, and at the same time is exposed to
a chance of dying which is the same at all ages. In
such an inconceivable population the age of the
individuals obviously has no significance. In practice,
a population has a certain age schedule both of
fecundity and of mortality. Now a population with
constant age schedules of fecundity and mortality,
which is multiplying in an unlimited environment,
will gradually assume a fixed age distribution known
as the stable age distribution' (Lotka, I925, p. I IO).
When this age distribution is established the population will increase at a rate dN/dt = rN. Thus the
parameter r refers to a population with a stable age
distribution. The consideration of rates of increase
in terms of the stable age distribution was one of the
most important advances in vital statistics. In any
other sort of population the rate of increase varies
with time until a stable age distribution is assumed.
There is, for example, no simple answer to the
question: what is the rate of increase of x newly
emerged adult insects in an unlimited environment?
The rate will vary with time as immature stages are
produced until the population has a stable age distribution. The rate of increase in the first generation
might be given, but that is a figure of limited value.
On the other hand, the maximum rate that it can ever
maintain over an indefinite period of time is given by
the rate of increase in a population of stable age distribution. That rate is therefore the true intrinsic
capacity of the organism to increase. Thompson
rejected the use of the exponential formula in
(193I)
the study of insect populations in preference for
a method of dealing with the rate,of increase as
a 'discontinuous phenomenon'. His paper should
be consulted for the reasons why he considers a single
index unsatisfactory in relation to the particular
problems with which he was concerned.
If the 'biotic potential' of Chapman is to be given
quantitative expression in a single index, the parameter r would seem to be the best measure to adopt,
since it gives the intrinsic capacity of the animal to
increase in an unlimited environment.* But neither
'biotic potential' nor 'true rate of natural increase'
can be regarded as satisfactory descriptive titles. The
word 'potential' has physical connotations which
are not particularly appropriate when applied to
organisms. There is a sense in which it might be
better used with reference to the environment rather
than the organism. Contrary to what it seems to
imply, the 'true rate of natural increase' does not
describe the actual rate of increase of a population at
a particular point in time, unless the age distribution
of that population happens to be stable. But it does
define the intrinsic capacityof that population, with its
given regime of fecundity and mortality, to increase.
This point is clearly made by Dublin & Lotka (I925).
More recently, Lotka (I945) has dropped the use of
'true rate of natural increase' for the more precise
'intrinsic rate of natural increase'. It would seem
desirable that students of populations should adopt
the same terminology, irrespective of the animals
concerned, and as 'intrinsic rate of natural increase'
is more truly descriptive of the parameter r than other
alternatives, its use is adopted in this paper.
The intrinsic rate of increase of a population may
be calculated from the age-specific fecundityt and
survival rates observed under defined environmental
conditions. For poikilothermic animals these rates
vary with physical factors of the environment such as
temperature and humidity. Furthermore, within any
* For a discussion of the relative merits of this and other
parameters in human demography reference should be
made to Lotka (945).
t Fecundity rate is used to denote the rate at which eggs
are laid by a female. Some eggs laid are infertile and so do
not hatch. The percentage 'infertility' is included in the
mortality rate of the egg stage. It is usual amongst
entomologists to denote the percentage of fertile eggs as
the 'fertility rate'. Demographers, on the other hand, use
'fertility rate' to denote the rate of live births. Since
'fertility rate' has this other usage in entomology the term
'fecundity rate' is used throughout this paper as synonymous with the 'fertility rate' of the demographers.
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L. C.
given combination of physical factors, the fecundity
and survival rates will vary with the density of the
animals. Hence it is possible to calculate an array of
values of r at different densities. But particular
significance attaches to the value of r when the
fecundity and survival rates are maximal, i.e. when
density is optimal, for this gives the maximum possible
rate of increase within the defined physical conditions.
Between the whole array of physical conditions in
which the animal can survive there is a zone where
fecundity and survival rates are greatest and where,
therefore, the intrinsic rate of increase will be greatest
too. The zone within which the intrinsic rate of
increase is a maximum may be referred to as the
optimum zone. This is an arbitrary use of the word
optimum and it does not imply that it is always to the
advantage of the animal to increase at the maximum
possible rate. The maximum intrinsic rate of increase
under given physical conditions has importance from
two points of view. It has a theoretical value, since
it is the parameter which necessarily enters many
equations in population mathematics (cf. Lotka,
1925;
Volterra,1931; Gause, 1934; Crombie,I945).
It also has practical significance. The range of
temperature and moisture within which the insect
can multiply is defined most precisely by that range
within which the parameter exceeds zero. This will
define the maximum possible range. In nature the
range of physical conditions within which the species
may be found to multiply may be less, since it is
possible that effects of density and interspecific competition may reduce this range, and also the range
of the optimum zone. These considerations are, however, beyond the scope of this paper; some discussion
of them will be found in a review paper by Crombie
( I947) l
There are some important differences in the
orientation with which the demographer and the
student of insect populations face their problems.
In human populations the parameter r varies in
different civilizations and at different times in one
civilization, depending upon customs, sanitation and
other factors which alter mortality and fecundity
rates. The maximum possible value of r does not
enter into most demographic studies. In a population
which is growing logistically the initial rate of increase
is theoretically the maximum intrinsic rate of increase,
and this latter value can be determined indirectly by
calculating the appropriate logistic curve. Lotka
(I927)
has done this for a humanpopulationand so
arrived at an estimate of a physiological maximum for
man. This has theoretical interest only. In insect
populations, on the other hand, the maximum value
for the intrinsic rate of increase does assume considerable theoretical and practical significance, as has
already been pointed out. The entomologist can
readily determine the maximum values and this is his
BIRCH
I7
obvious starting-point. But the determination of r at
different stages in the population history of an insect,
whether in an experimental population or in the field,
offers many practical difficulties which have not yet
been surmounted for any single species. The values
which the entomologist has difficulty in determining
are those which are most readily obtained for human
populations. The crude birth-rates and crude deathrates of the population at specific stages in its history
are precisely those indices with which the demographer works. His census data provides him with
the actual age distribution which is something not
known empirically for a single insect species. He can
have a knowledge of age distribution even at intercensal periods, and under civilized conditions he can
also determine the age-specific rates of fecundity and
mortality which were in operation during any
particular year. In insect populations this is at
present impossible; one can only keep a number of
individuals under specified conditions and determine
their age-specific rates of fecundity and survival, and
from these data r can be calculated.
The fact that populations in nature may not realize
the maximum value of their intrinsic rate of natural
increase, does not negate the utility of this parameter
either from a theoretical or a practical point of view.
Having determined this parameter, the next logical
step is to find out the extent to which this rate of
increase is realized in nature. It is conceivable that
some species, such as those which infest stored wheat
or flour, may increase exponentially when liberated
in vast quantities of these foodstuffs. This would
imply that the insects could move out of the area in
which they were multiplying with sufficient speed to
escape density effects and that they had no gregarious
tendencies. An exponential rate of increase may also
occur in temperate climates in some plant-feeding
species which only multiply in a short period of the
year in the spring. In seasons with abundant plant
growth the insect population may be far from
approaching any limitation in the resources of the
environment before the onset of summer retards the
rate of increase. The population counts of Thrips
imaginisin some favourable seasons in South Australia
suggest such a picture (Davidson & Andrewartha,
1948).
3. CALCULATION OF THE INTRINSIC
RATE OF NATURAL INCREASE
(a) Experimental data required
The calculation of r is based on the female population;
the primary data required being as follows:
The female life table giving the probability at
(i)
birth of being alive at age x. This is usually designated 1, (lo= I).
(2) The age-specific fecundity table giving the
J. Anim. Ecol. 17
2
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The intrinsicrate of natural increaseof an insectpopulation
mean number of female offspring produced in a unit
of time by a female aged x. This is designated m,.
In the calculation of the stable age distribution the
age-specific survival rates (Ix) of both the immature
stages and the reproductive stages are required. For
the calculation of r the life table of the adult and only
the total survival of the immature stages (irrespective
of age) are needed. In practice, the age-specific
fecundity rates mx will be established for some convenient interval of age, such as a week. If N eggs are
laid per female alive between the ages x to x + i in
the unit of time chosen, then m, simply equals ,N
when sex ratio is unity. It is assumed that this value
occurs at the mid-point of the age group.
A numerical example is worked out for the rice
weevil Calandra (Sitophilus) oryzae (L.) living under
optimum conditions (290 C. in wheat of I4%
moisture content). Data for the rates of development
and survival of the immature stages, and the agespecific fecundity rates were obtained from Birch
(Ig45 a, b). The life table of adult females has not
been determined experimentally, only the mean
length of adult life being known. However, an
estimate was obtained for purposes of these calculations by adapting the known life table of Tribolium
confusum Duval (Pearl, Park & Miner, I941) to
Calandra oryzae, making the necessary reduction in
the time scale. Since the mean length of life of
Triboliumconfusumin this life table was I98 days and
the mean length of life of Calandra oryzae at 290 was
84 days, one 'Calandra day' has been taken as
equivalent to 2-35 'Tribolium days'. To this extent
the example worked out is artificial, but, for reasons
which will become evident later in the paper, it is
unlikely that the error so introduced in the estimate
of r is of much significance.
Before proceeding to outline direct methods of
estimating r two other parameters must first be
mentioned: the net reproduction rate and the mean
length of a generation.
(b) The net reproductionrate
This is the rate of multiplication in one generation
(Lotka, 1945) and is best expressed as the ratio of
total female births in two successive generations. This
we shall call Ro and so follow the symbolism of the
demographers. Ro is determined from age-specific
fecundity and survival rates and is defined as
00
RO= lfmxdx,
where 1, and ms, are as already defined.
The method of calculating Ro is set out in Table i.
The values of 4, are taken at the mid-point of each age
group and age is given from the time the egg is laid.
Since the survival rate of the immature stages was
o9go the life table of adults reckoned from 'birth',
i.e. oviposition, was the product: l, for adults x o 9o.
Development from the egg to emergence of the adult
from the grain lasts 28 days and 4 5 weeks is the midpoint of the first week of egg laying. The product
la,mxis obtained for each age group and the sum of
these products El,m^ is the value Ro. In this
particular example Ro = I 13-6. Thus a population of
Calandra oryzae at 290 will multiply II 3-6 times in
each generation.
Table I. Showing the life table (for ovipositionspan)
age-specificfecundity rates and the method of calculating the net reproduction rate (RO)for Calandra
oryzae at 29? in wheat of 14% moisture content.
Sex ratio is equal
Pivotal age
in weeks
(X.
45
55o583
(M.)
(I)
o-87
20-0
23-0
6.5
o-8i
15.0
75
8-5
o-8o
12.5
o079
12.5
9 5
10-5
0?77
0 74
I4-0
12.5
115
o-66
14-5
I2-5
0 59
II0
(xx
17.400
I9-090
I2-150
10-000
9 875
I0-780
9-250
I3.5
0-52
9-5
9 570
6-490
4 940
I4-5
15.5
0?45
2.5
2-5
I-I25
0-900
I6.5
0-29
2.5
o-8oo
I7.5
0-25
40 1?000
I8-5
O-I09
I0
0-36
O-I90
Ro= 113-560
The comparison of two or more populations by
means of their net reproduction rates may be quite
misleading unless the mean lengths of the generations are the same. Two or more populations may
have the same net reproduction rate but their intrinsic
rates of increase may be quite different because of
different lengths of their generations. Consider, for
example, the effect of moving the lXmZcolumn in
Table i up or down by a unit of age, Ro remains the
same but it is obvious that the generation times are
now very different. For these reasons the parameter
Ro has limited value and it must always be considered
in relation to the length of the generation (T).
(c) The mean length of a generation
The relation between numbers and time in a
population growing exponentially is given by
N_ = NoerT.
When T= the mean length of a generation, then from
the definition of net reproduction rate NT/No =Ro
hence
Ro = e",
and
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T_ loge Ro
r
L. C.
I9
BIRCH
tions to the rigorous procedures are justified in so far
as the determination of the primary data which enter
the above formula is of course subject to considerable
error, arising from the normal variation in the
organisms and conditions to which they are subjected in the experiments. It was considered that an
estimate of r, calculated to the second decimal place,
was sufficient in these circumstances. The following
approximate method was therefore adopted. It has
1925).
For approximatepurposesthereforeit may the merit of being both simple and fast.
As an approximation we may write
be defined as
_2xl mM
It follows that an accurate estimate of the mean
length of a generation cannot be obtained until the
value of r is known. For many purposes, however,
an approximate estimate of T which can be calculated
independently of r may be of use. Thus, although
oviposition by the female is extended over a period
of time, it may be considered as concentrated for each
generation at one point of time, successive generations being spaced T units apart (Dublin & Lotka,
T=
e r"lxm.
.
S
We may thus consider the figures for the product
lxmxgiven in the last column of Table i as a frequency
distribution of which the individual items are each
concentrated at the mid-point of each age group. The
mean of this distribution is the approximate value of
T. In this particular example
T=943099=8-3
II3-56
weeks.
If this were an accurate estimate of T we could
proceed to calculate the value of r since, from the
above equation relating Ro, r and T, we have
loge Ro
T
9 II356 =o57
per head per week.
It will become evident in what follows that this is an
underestimate of r owing to the approximate estimate
of T. The procedure does, however, serve to illustrate
the nature of the parameter, and in some cases where
r is small it may be a sufficiently accurate means of
calculation (cf. for example, Dublin & Lotka, 1925).
We shall proceed in the next section to an accurate
method for the calculation of r.
(d) The calculation of 'r'
A population with constant age schedules of
fecundity and mortality will gradually approach a
fixed form of age distribution known as the stable age
distribution (p. i6). Once this is established the
population increases at a rate dN/dt=rN and the
value of r may be calculated from the equation
fe-rxlIxmxdx=
I.
For the derivation of this formula reference must be
made to Lotka (I925) and the bibliography therein.
The usual methods of calculation may be found in
Dublin & Lotka (I925, Appendix) or Lotka (1939,
p. 68 et seq.). For high values of r, these methods may
not be particularly satisfactory (Leslie & Ranson,
I940;
Leslie, 1945, Appendix), and the computations,
moreover, become very tedious. Some approxima-
= I.
Here x is taken to be the mid-point of each age group
and the summation is carried out over all age groups
for which m > o. A number of trial values are now
substituted in this equation, in each case calculating
a series of values e-rx and multiplying them by the
appropriate lm. values for each age group. By
graphing these trial values of r against the corresponding summation values of the left-hand side of
the above expression, we may find the value of r which
e-rl
will make
m. +I.
The whole procedure is greatly simplified by the use
of 4-figure tables for powers of e (e.g.Milne-Thomson
& Comrie I944, Table 9). Since these tables only
give the values of e-- at intervals of o-oi in the
argument x up to e?6, it may be convenient to
multiply both sides of the equation by a factor ek in
order to work with powers of e which lie in the more
detailed parts of the table. Thus, in the present
example, k was taken as 7:
e7e
EerXlmx = e7
Ee7-rT.lmx=
I097.
A value of r was now sought which would make the
left-hand side of this expression equal to 1097. The
actual process of carrying out this simple computation is exemplified in Table 2. The summation of the
expression is not carried beyond the age group
centred at I3-5 because of the negligible contribution
of the older age groups. Ithas already been mentioned
that r is an infinitesimal rate of increase not to be
confused with a finite rate of increase Awhich equals
antiloge r. In this particular example r = o-76 and
In other words the
A therefore has a value 2-14.
population will multiply 2- I4 times per week.
By reference to Table 2 it is clear that the relative
weights with which the different age groups contribute to the value of r are given by the values lxmxe7-rx
at each age group. It is of particular interest to
observe the relation between values at successive age
intervals (Table 3). The value of r is 56o%accounted
for by the first week of adult life. The first 2 weeks
combined contribute 85 % towards the final value and
the first 3 weeks combined total 94%. The I3-5th
week, on the other hand, contributes 0-02 %. It
2-2
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The intrinsicrate of natural increaseof an insectpopulation
20
should not be inferred that adults I 3-5 weeks old are of
no importance since their eggs will eventually give rise
to adults in the productive age categories. The biological significance of Table 3 is that the intrinsic rate
of increase is determined to a much greater extent by
the rate of oviposition in the first couple of weeks of
adult life than by the total number of eggs laid in the
life span of the adult, even although only 27 % of the
total number of eggs are laid in the first 2 weeks. With
Table 2. Showing the method of calculating r for
Calandra oryzae at 29? by trial and error substitutions in the expression
Pivotal
age group
lxm.
(x)
45
555
I7-400
9-090
12-150
10 000
9 875
I0-780
6.5
75
8.5
9.5
I0 5
II*5
Ee7-rxlxm,M=
1097
r = o-76
-x
9-250
7-rx
e7-rx
3-58
2-82
2-o6
130
0 54
-0-22
-o-98
9 570
I-74
12.5
6-490
- 2-50
I3 5
4-940
r=o 77
e7-rv
7-rx
35 87
3 53
I6-78
7-846
3-669
1-7I6
o-8025
0?3753
0 I755
2.76
I-99
3-26
1-22
0?45
-0 32
-1-09
- i86
34.12
I5-80
7316
3-387
15683
0-726I
0-3362
0-1557
o-o821
-2-62
00728
0?0384
-339
0?0337
13-5
i e7-rxlmx
=
iio8
I047
follows that in determining oviposition rates experimentally, the rates in early adult life should be found
with the greatest accuracy. Of corresponding importance is the accurate determination of the pivotal
age for the first age category in which eggs are laid. In
the example being cited an error of half a week causes
an error of 8 % in the estimate of r.
The calculations were repeated ignoring the adult
life table. The value of r was then 0o77. Since the
imposition of an adult life table onlymakes a difference
of i % in the value of r it is evident that the life table
is of little importance in this example. This is due to
the fact already noted that the major contribution to
the value of r is made by adults in early life, and during
early adult life survival rate is at a maximum. The life
table may assume quite a different importance in a
species with a different type of age schedule of
fecundity or when the value of r is lower.
4. THE STABLE AGE DISTRIBUTION
With a knowledge of the intrinsic rate of increase and
the life table it is possible to calculate the stable age
distribution and the stable female birth-rate of the
population. Thus if c$ is the proportion of the stable
population aged between x and x + dx, and b is the
instantaneous birth-rate
Cx= be-rxlx,
4-5
r lies between 076 and 0-77 and by graphicalinterpretation =
0-762.
Table 3. The contribution of each age group to the
value of r when r =o76
Pivotal age
group
(x)
Percentage
contribution of
lxmxe7-rx
each age group
4 5
624-I
56 33
5.5
28-9I
86o
7.5
320-3
95 3
36 7
8.5
I70
6-5
3.31
I153
95
87
0o78
I0 5
1 1I5
I2-5
3 5
I.7
05
0-32
I3.5
0-2
0-02
iio8*o
OI5
005
ioo0oo
each successive week, eggs laid make a lessened contribution to the value of r. In this particular case this
can be expressed by stating that for each egg laid in
the first week of adult life it would require 2 I times
as many in the second week to make the same contribution to the value of r, (2 I)2 in the third week and
(2 i)n-1 in the nth week. The ratio 2zI: I is the ratio
between successive weighting values e7-,x (per egg)
in Table 2. The importance of the first few weeks is
further intensified by the fact that egg laying is at
a maximum then. From these considerations it
.0
and
i/b =
e-rxlxdx.
For the usual methods of computation reference
should be made again to Dublin & Lotka (I925).
Mr Leslie has, however, pointed out to me another
method of calculation which saves much of the
numerical integration involved in the more usual
methods. At the same time it is sufficiently accurate
for our present purpose. If at time t we consider
a stable population consisting of Nt individuals, and
if during the interval of time t to t + i there are Bt
female births, we may define a birth-rate
P=BBtNt.
Then if we define for the given life table (lx) the series
of values L., by the relationship Lx =
rx+1
lxdx (the
x
stationary or 'life table' age distribution of the
actuary),* the proportion (px) of individuals aged
between x and x + i in the stable population is given by
Px= pLXe-r(x+l)
m
I
X=0
Lxe-r(X+
where x = m to m + i is the last age group considered
in the complete life table age distribution. It will be
noticed that the life table (lx values) for the complete
* For a discussion of Lx see Dublin & Lotka (1936).
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L. C.
age span of the species are required for the computation of px and I. But where r is high it will be found
that for the older age groups the terms Lie-r(+l) are
so small and contribute so little to the value of f that
they can be neglected.
The calculations involved are quite simple and are
illustrated in the following example for Calandra
oryzae at 290 (Table 4). Actually, in the present
example, instead of calculating the values of L,, the
values of 14were taken at the mid-points of each age
group. This was considered sufficiently accurate in
the present instance. It should also be pointed out
that whereas only the total mortality of immature
stages was required in the calculation of r, the age
specific mortality of the immature stages is needed
Table 4. Calculation of the stable age distribution of
Calandra oryzae at 290 when r = 0o76
Age
group
Percentage
distribution
(x)
L
0I-
0-95
0o4677
0-4443I50
54 740j
090
0-I968300
24-249
2 -
0o90
0-2I87
0-I0228
0o0920520
II34I
34567-
0o90
o-87
o-83
o-8i
o-8o
89 IO-
0-79
0-77
e-r(x+l)
Lx e-r(x+l)
0-04783
0o0430470
5-304J
0o02237
0-01946I9
2-398
0-OI046
0-00489
o-oo868i8
0-003960g
0-488
0002243
0-00I070
0-0017944
0-22I
o-ooo8453
01I04
0-000500
0-0003850
0?047
0-022
0-74
o-66
0o000239
0o0001769
O-OOOIIO
0-0000726
I3-
0-59
0-52
0 000051
0o000024
0-000OI25
14-
0-45
I
O-OOOOI
0-0000050
II12-
IoopLxe-r(x+l)
0-000030I
If/f=
o*8 II6704
95.5 %
total
will
be 0o95
1-070
whole insect population. Methods of sampling are
required which will take account of the immature
stages hidden inside the grains, such, for example, as
the 'carbon dioxide index' developed by Howe &
Oxley (I944). The nature of this stable age distribution has a bearing on another practical problem. It
provides further evidence to that developed from
a practical approach (Birch, I946) as to how it is
possible for C. oryzae to cause heating in vast bulks
of wheat, when only a small density of adult insects
is observed. It is not an unreasonable supposition
that the initial rate of increase of insects in bulks
of wheat may approach the maximum intrinsic rate
of increase and therefore that the age distribution
may approach the stable form. Nothing, however, is
known about the actual age distribution in nature at
this stage of an infestation.
5. THE INSTANTANEOUS
BIRTH-RATE
AND DEATH-RATE
We have already defined a birth-rate g by the
expression
m
LXe-r(x+l)
This is not, however, the same as the instantaneous
birth-rate (b) where r = b-d.
In personal communications Mr Leslie has provided me with the
following relationship between these two birth-rates.
rg
er-I
adults
O-OOI
100*000
o9go
E
X=O
4-5 %
total
o0oog
0-004
0-002
and thereafter
2I
I/-
immature
stages
for the calculation of the stable age distribution. In
this example the total mortality of the immature
stages was iO %-and 98 % of this mortality occurred
in the first week of larval life (Birch, I 945 d). Hence
-the approximate value of Lx for the mid-point of the
-first week
BIRCH
for
successive weeks of the larval and pupal period
(column 2, Table 4). The stable age distribution is
shown in the fifth column of Table 4. This column
simply expresses the fourth column of figures as
percentages.
It is of particular interest to note the
high proportion of immature stages (95-5 %) in this
-theoretical population. This is associated with the
high value of the intrinsic rate of natural increase.
It emphasizes a point of practical importance in
estimating the abundance of insects such as C. oryzae
and other pests of stored products. The number of
adults found in a sample of wheat may be quite
-a misleading representation of the true size of the
Thus, in the example for C. oryzae, we have
I/p=o.8II67 (Table 4), r=0o76 and thus b=o-82
and the difference between r and b is the instantaneous
death-rate (d) = o-o6.
The instantaneous birth-rate and death-rate are
widely used by students of human populations. The
insect ecologist is more likely to find greater use for
the finite rate of increase A (natural antiloge).
6. THE EFFECT OF TEMPERATURE
ON 'r'
As an illustration of the way in which the value of
r varies with temperature and the corresponding
changes in rateof development, survival and fecundity,
an estimate of r for C. oryzae has been made for two
temperatures (230 and 33.50 C.) on either side of the
optimum
(290).
The
span of adult life at 230 is
about the same as at the optimum 290 and so the same
life table has been applied. Even although the egg
laying is more evenly distributed throughout adult
life the life table makes little difference to the value of
r. Furthermore, the first 2 weeks of adult life carry
a weight of 59 % of the total weight of all age groups
in the determination of the value of r. For every egg
laid in the first week of adult life it would require I S
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The intrinsicrate of natural increaseof an insectpopulation
22
times as many eggs in the second week to make the
same contribution to the value of r, and 2-3 times as
many eggs in the third week to have the same effect.
The relative weight of each week decreases less with
successive weeks at 23' than at 29?. This is associated
with the lower oviposition rates and the longer
duration of the immature stages at 230.
At 33.50 egg laying ceases after the fourth week of
adult life, the mortality of adults during these 4 weeks
is not high and so the estimate of r obtained without
a life table may not be very different from the true
value.
Table 5. Showing the values of 1,, mi, and the estimate
of r for Calandra oryzae at 23' and 33 50
230
335
Pivotal age
in weeks
(X)
Pivotal age
in weeks
lsc
mx
(x)
lX
M.
05)
0-5
Immature0-90
stages
Imature
stages
-
6~~~~~~.
0o25
7-5
8.5
o-87
o083
II-0
10o5
025
0 25
6-o
35
95
o08I
II.5
II.5
0o25
3-0
I2-0
I2-5
0-25
I-0
II1-5
o-8o
?079
I2.5
0?77
I3-0
13-5
I4.5
0-74
II5
Ii0
1.05
9-0
115
r=oI
15-5
o-66
o 6o
I6-5
0052
II-0
17-5
0?45
I2
I8-5
o036
10 5
II.5
40
19.5
0-29
205
025
0-I9
2I*5
9-5
2
per head per week
I0o0
5
2-0
With adult life table
r = 0-43 per head per week.
Without adult life table r = o 44
7. DISCUSSION
In order for a species to survive in a particular
environment it may need to have evolved a certain
minimum value for its intrinsic rate of natural increase. If its rate of increase is less than this it may
succumb in the struggle for existence. It does not
necessarily follow that the higher the intrinsic rate of
increase the more successful will the species be.
Evolution may operate to select species with an intrinsic rate of increase which is both large enough to
enable them to compete successfullywith other species
and small enough to prevent a rate of multiplication
whicn would exhaust the food supply in the environment. Whatever is the minimum necessary value of
'r' it could be attained along more than one route,
since r has a number of component variables; the
length of development of the immature stages, the
survival rate of the immature stages, the adult life
table and the age-specific fecundity schedule. These
components enter into the value of r with various
weights, and it is suggested in the discussion which
follows that a knowledge of their relative contributions
may provide a clue to the significance of the life
patterns characteristic of different species. There is
clearly a pattern in the seasonal environment too,
which must be considered at the same time. A hot dry
period, for example, may necessitate a prolonged egg
stage. In an environment which has relatively uniform physical conditions all the year round, these
complicating factors are at a minimum, e.g. a tropical
forest or the micro-environment of a stack of wheat.
(i)
Consider first the length of the immature stages
(non-reproductive period) in relation to the span of
egg laying and the age schedule of fecundity. The
earlier an egg is laid in the life of the insect the greater
is the contribution of that particular egg to the value
of r. In illustration we may consider the age schedule
of fecundity for C. oryzae at 290. Since over 95 %
of the value of r is determined by the eggs laid in the
first 4 weeks of adult life (Table 3) we can, for purposes of illustration, ignore the remaining period.
At 29? the immature stages of C. oryzae last 4 weeks
and the maximum rate of egg laying is 46 eggs per
week (Table i). Now the same value of r (o-76) is
given in a number of imaginary life cycles by reducing
the length of the immature stages along with a reduction in the rate of egg laying and alternatively by
increasing both the length of the immature stages and
the number of eggs laid (ordinary figures, Table 6).
In Table 6 the age schedule of fecundity is kept proportionate in each case. In the extreme examples if
the immature stages could develop in a week, an
oviposition maximum of 5 eggs per week would give
the same rate of increase as the imaginary insect which
took 6 weeks to develop and had an oviposition
maximum of 204 eggs per week. The imaginary life
cycles have been calculated from the ratio (2-I :I)
for successive weighting values (e7-rx) in Table 2.
The question might now be asked, what determines
the particular combination which the species happens
to possess? In the specific example in question, if the
larva took 6 weeks to develop the adult would need to
lay 2oo eggs per week. But it now becomes necessary
to consider the behaviour pattern, for C. oryzae bores
a hole in the grain of wheat for every egg which is laid.
The whole process of boring and egg laying occupies
about i hr. per egg. So that with this particularmode
of behaviour 24 eggs per day would be an absolute
maximum. There must of course also be some physiological limit to egg production. For a larger insect
the physiologicai limit might be less restrictive provided that the size of the egg does not increase
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L. C.
BIRCH
23
The relative advantage of this type of fecundity
schedule is less, the smaller the value of r. At 23? the
value of r for C. oryzae is 0-43 and the eggs laid in the
first week of adult life are worth (1-5)n-1 eggs in the
nth week (compare this with the value of (z i)n-I
when r= 076). The actual oviposition time curve
at 230 has no distinct peak as at 290 (Tables i
and 5).
There is a wide variation in the nature of the age
schedule of fecundity amongst different species of
insects with perhaps the tsetse fly and the lucerne
flea illustrating contrasting extremes. Whereas tsetse
flies (Glossina)deposit single larvae spaced at intervals
of time, the' lucerne flea' (Smynthurusviridis) deposits
its eggs in one or two batches of as many as I20 at
proportionately with the size of the insect. In considering this possibility, ecological considerations are
important, for C. oryzae is adapted to complete its
development within a grain of wheat and a size limit
is set by the length of the grain. There is, in fact,
a strain which is found in maize kernels in Australia
and this is considerably larger than the so-called
'small strain' (Birch, I944). Furthermore, the larger
insect would probably require a longer time to complete development (which is actually the relationship
observed between the small and large strains) and
this would operate to reduce the value of r. In considering the possibilities in the opposite direction,
there is obviously a limit below which the length of
development could not be reduced any further.
A species of smaller size could doubtless develop in
a shorter time and on this merit might be a more
successful mutation. But the question then arises
whether a smaller species could command muscles
and mandibles of sufficient strength to chew whole
a time (Maclagan,I932).
The particularadvantage
of this mode of oviposition must be tremendous, and
is probably responsible in part for the great abundance
of this collembolan and possibly other members of
the same order, which, as a whole, are among the most
Table 6. Showing the actual relation between the length of the immature stages and the age schedule of
fecundity for Calandra oryzae at 29? (black figures) and some theoretical possibilities which would give the
same intrinsic rate of increase (r = 076). The length of the immature stages is shown in the left of the table;
figures in the body of the table are numberof eggsper week
Pivotal age in weeks
0-5
1I5
2-5
week
2weeks
4
5
I
3-5
3
9
3 weeks
5-5
10
4-5
3
7
I9
22
I4
40
4 weeks
5 weeks
6 weeks
8-5
I2
-
-
46
30
25
-
-
141
84
97
63
53
-
297
564
6
Age schedule of fecundity of Calandra oryzae
at 290
Weeks
Actual
I
40
2
3
4
46
30
Imaginary
7I
0
0
25
0
Total
141
71
204
I32
Total
I5
32
67
-
II7
grain. Grain-feeding species of beetles which are
smaller than C. oryzae are in fact scavengers rather
than feeders on sound grain. Thus it would seem that
a balance is struck somewhere between the minimum
time necessary for development and the maximum
possible rate of egg laying, and this is conditioned by
the behaviour pattern of the insect and the particular
ecology of its environment.
For a maximum value of r the optimum age schedule
of fecundity is one which has an early maximum. In
an imaginary schedule for C. oryzae a concentration
of 7I eggs in the first week of egg laying would give
the same value of r (o-76) as 14i eggs distributed over
4 weeks.
95
7-5
6-5
III
abundant insects in nature. This is of course speculative and much more information is required before
any generalizations can be made. Another interesting
category are the social insects, since only one female
of the population (in termites and the hive-bee) or
a few (in social wasps) are reproductives. A theoretical
consideration of the relative merits of one queen and
many queens might throw more light on the evolution
of these systems, especially as they relate to differences
in behaviour.
The relation between the length of the pre-reproductive stages -and the nature of the age-fecundity
schedule is in part dependent upon the nature of the
seasonal changes in the environment. Life histories
may be timed so that the reproductive and feeding
stages coincide with the least hostile season of the
year. Diapause, aestivation and hibernation are some
of the adaptations which ensure this. They have
particular significance too in determining the age distribution of the initial population in the reproductive
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24
The intrinsicrate of natural increaseof an insectpopulation
season. Consideration of this is left to a later section
of the discussion.
(2) For the calculation of the maximum intrinsic
rate of increase the life table of the species from
deposition of the egg (or larva) to the end of egglaying life in the adult must be known. The startingpoint of this life table is thus the stage which corresponds to the point of 'birth'. Deevey (I 947) has
noted that the point of universal biological equivalence
for animals is doubtless fertilization of the ovum. But,
from the point of view of the number of animals in the
population and for purposes of calculating r, a knowledge of pre-birth mortality is not required. For the
calculation of r the life table beyond the end of
reproductive life has no significance, but a knowledge
of age-specific post-reproductive survival until the
point of death is needed, on the other hand, for the
calculation of the stable age distribution and the
instantaneous birth-rate. This is evident from a consideration of the method of calculation shown in
Table 4. The post-reproductive life assumes negligible
significance in this particular example, but its importance in such calculations increases as r approaches
a value of zero.
The relative importance of the survival pattern
(i.e. the shape of the 1, curve) in determining the value
of r is itself a function of r. When r is small its value
may be dependent to a significant extent on the oviposition in late adult life, when it is large it is mostly
determined by the oviposition rates of adults in early
adult life. When the intrinsic rate of increase is high and
the life table of the adult follows the typical diagonal
pattern (e.g. Pearl et al. I94i) with no high mortality
in early adult life, consideration of the adult life table
is of little importance in calculating r. This is because
survival rate is high in the ages which contribute most
to the value of r. In species which have a low intrinsic
rate of increase the life table may assume more
significance in determining the value of that rate of
increase. More data are required before the importance of this point can be established. The pattern
of survival which gives a maximum value of r has its
maxima in the pre-reproductive and early reproductive stages. A knowledge of total survival of the
immature stages is of course essential in all cases.
More attention might well be given by entomologists
to securing life table data than has been given in the
past. Without it no true picture of the intrinsic rate
of increase can be obtained.
(3) There remains to be considered the age distribution of the population in relation to its capacity
to increase in numbers. In a population in an unlimited environment the stable age distribution is the
only one which gives an unvarying value of r. For
this reason the stable age distribution is the only
sound basis on which to make comparisons between
different values for rates of increase (whether between
different species or one species under different
physical conditions). The actual age distribution of
a population in nature may be quite different and its
consideration is of importance in determining the
initial advantage one form of distribution has over
another. In an unlimited environment these initial
differences in age composition are eventually ironed
out. A population which initiates from a number of
adults at the peak age of egg la'yingclearly has a higher
initial rate of increase than one which starts from the
same number in all differeit stages of development.
These considerations may be of most importance
in temperate climates where there is a definite seasonal
occurrence of active stages. The stage in which the
insect overwinters or oversummers will determine
the age distribution of the population which initiates
the seasonal increase in the spring or in the autumn
(whichever the case may be). The pea weevil, Bruchus
pisorum, in California hibernates as an adult. With
the first warm days in the spring the adults leave
their overwintering quarters under bark and fly into
the pea fields (Brindley, Chamberlin & Hinman,
I946). Following a meal of pollen they commence
oviposition on the pea crops. This mode of initiating
the spring population would be far more effective
than one which started with the same number of
insects in the egg stage. The overwintering adults
begin their reproductive life at a much later age than
the adults in the next generation. It would be of
interest to know whether the age schedule of fecundity
(taking the commencement of egg laying as zero age)
is the same for both generations. This is a point which
does not appear to have been investigated for insects
which hibernate as adults. It is clearly of much
importance in determining the intrinsic rate of increase of successive generations.
Overwintering as pupae must theoretically rank
as the second most effective age distribution for
initiating spring increase. Many species which overwinter as pupae would have a higher mortality if they
overwintered as adults. The corn-ear worm, Heliothis
armigera, for example, can hardly be conceived as
overwintering as an adult moth in the North American
corn belt. In the northern part of this belt even the
pupae which are protected in the soil are unable to
survive the winter. Recolonization evidently takes
place each year from the warmer south (Haseman,
1931).
Overwintering in the egg stage (in hibernation
or in diapause) is common in insects. Here again
it is difficult to imagine the other stages of these
orders as successfully hibernating. The grasshopper,
Austroicetes cruciata (Andrewartha, 1944), and the
majority of aphids are examples of this. A minority
of aphid species are, however, able to overwinter as
apterae by finding protection in leaf axils and similar
niches (Theobald, 1926), some others are enabled to
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L. C.
survive as adults by virtue of their symbiosis with ants
(Cutright, I925).
BIRCH
25
concept for the study of insect populations. It is sug-
The relatively vulnerableaphids gested thatfor the sake of uniformity of terminology in
The preceding examples illustrate how the age
distributions of initiating populations vary in seasonal
species. This depends on the nature of the overwintering or oversummering stage. The particular
stage may have been selected in nature not only by
virtue of its resistance to unfavourable physical conditions but also in relation to its merits in initiating
rapid establishment of a population in the spring and
autumn. The calculation of the initial and subsequent
rates of increase of populations with these different
types of age distribution is considerably more complicated than the calculation of intrinsic rates of
increase for populations with stable age distributions.
This problem is not dealt with in this paper and the
reader is referred to Leslie (I945, p. 207 et seq.)
for an outline of the principles involved in such
calculations.
The length of the developmental stages, the age
schedule of fecundity, the life table of the species and
the age distribution of initiating populations present
a pattern which has adaptive significance for the
species. The analytic study of the intrinsic rate of
increase of a species (as exemplified by Calandra
oryzae) may throw light on the evolutionary significance of the life pattern of different species. Such
a study must necessarily be related to the behaviour
pattern of the insect and the type of environment it
lives in. Nor can the importance of effects of density
and competition be overlooked. These are, of course,
studies in themselves beyond the scope of this paper.
population biologyand for precision of definition, that
the term 'intrinsic rate of natural increase' might be
considered more appropriate than an alternative term
'biotic-potential' which is more frequently used in
relation to insect populations. The intrinsic rate of
natural increase is defined as the exponent 'r' in the
exponential equation for population increase in an
unlimited environment. The rate of increase of such
a population is given by dN/dt = rN. The parameter
r refers to the rate of increase of a population with
a certain fixed age distribution known as the stable
age distribution. Both the intrinsic rate of natural
increase and the stable age distribution may be
calculated from the age-specific survival rates (life
table) and age-specific fecundity rates. The methods
of calculation are exemplified with data for the rice
weevil, Calandra oryzae (L.), and some adapted from
the flour beetle, Tribolium confusum Duval. It is
shown in this example that the intrinsic rate of natural
increase is determined to a much greater extent by
the rate of oviposition in the first z weeks of adult life
than by the total number of eggs laid in the entire life
time. The oviposition rates in the first z weeks account
for 85 % of the value of r whereas only 27 % of the
total number of eggs are laid in that time. With each
successive week in the life of the adult, eggs laid make
a lessened contribution to the value of r. The methods
of calculation of r provide a means of determining
the extent to which the various components-the life
table, the fecundity table and the length of the prereproductive stages-enter into the value of r. It is
suggested that analyses of this sort may provide a clue
to the life patterns characteristic of different species.
The importance of the age distribution of populations which initiate seasonal increase in the autumn
and spring is discussed. These age distributions
depend on the nature of the overwintering or oversummering stage. It is suggested that this particular
stage, whether it be adult, larva, pupa or egg, has been
selected by virtue not only of its resistance to the
unfavourable season, but also in relation to its merits
in initiating rapid establishment of a population in
the succeeding season.
It is shown how the value of r for Calandra oryzae
varies with temperature. Four other parameters are
also defined: the net reproduction rate, the mean
length of a generation, the infinitesimal birth-rate
and the infinitesimal death-rate. The methods of
calculation of these parameters are also exemplified
with data for C. oryzae.
8. SUMMARY
The parameter known as the intrinsic rate of natural
increase, which was developed for demographic
analyses by A. J. Lotka, is introduced as a useful
9. ACKNOWLEDGEMENTS
Grateful acknowledgement is made to the Director
of the Bureau of Animal Population, Oxford University, Mr C. S. Elton, for the facilities of the Bureau
find protection in the nests of ants. The ants not only
carry them to their nests, but feed them during the
winter months and with the return of spring plant
them out on trees again!
Overwintering as nymphs or larvae is a rarer
phenomenon except with species which can feed and
grow at low temperatures and so are not hibernators.
The active stages of the lucerne flea, Smynthurus
viridis, for example, can be found in the winter in
Australia (Davidson, 1934). The seasonal cycle of the
reproducing population commences with the first
rains in autumn; this population being initiated with
oversummeringeggs. The eggs arethe onlystagewhich
are resistant to the dryness and high temperatures of
the summer months. Species of insects with hardy
adult stages like the weevil, Otiorrhynchus cribricollis (Andrewartha, I933), aestivate as adults. An
interesting case of a butterfly, Melitaea phaeton,
aestivating as a quarter-grown larva at the base of its
food'plantis describedby Hovanitz(I94I).
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26
The intrinsicrate of natural increaseof an insectpopulation
which were placed at the author's disposal during his
term there as a visiting worker. Mr Elton provided
much encouragement during the investigation. It is
a pleasure to acknowledge too the inspiration and help
of Mr P. H. Leslie of the Bureau of Animal Popula-
tion. His direction was indispensable in all mathematical and actuarial aspects of the paper and his
critical examination of the manuscript was much to
its advantage.
REFERENCES
'The bionomics of
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